Filtro de banda eliminada

Lemma 4.1. SupposeF : C1 →C2 is a map of filtered complexes that respects the filtrations. ThenF induces a map of spectral sequencesFn:IEn →IIEn, and ifFn is an isomorphism,Fm

is an isomorphism for all m≥n.

In [Lee05], for link diagramsLand ˜Lthat differ by a Reidemeister move, Lee provesKhLee(L)∼=

KhLee( ˜L) by creating a chain map betweenCKhLee(L) andCKhLee( ˜L). Thus to show that the spectral sequence is a link invariant, we just need to check that these chain maps respect the filtration, and that they induce isomorphisms on Khovanov homologyKh(L)∼=Kh( ˜L). For details, see [Ras10] [Lee05].

It also follows that each page of the spectral sequence is a link invariant by Theorem 4.4 below.

4.2

Equivariant Khovanov Homology

Recall that we described the Khovanov TQFT by

Z[

1

2]⊗ZHom(∅,O)/( = 0),

and the Lee TQFT by

Z[

1

2]⊗ZHom(∅,O)/( = 8).

The above Hom spaces are Z-modules, and then tensored withZ[12] to make 2 invertible to allow

the local relations inCob3_/l.

Now we choose to work over_Q[α], and we set the genus 3 surface to be 8α:

Definition 4.2. We define the ‘universal Khovanov homology’ or ‘equivariant Khovanov homology’1 Khα_{(T) of a tangleT to be the homology of the complex obtained via the TQFT}

Q[α]⊗ZHom(∅,O)/( = 8α).

The resulting complex by applying this TQFT to BN(T) will be denotedCKhα_{(T). Following}

Example 2.21, a surface carrying two dots is then equal toαtimes that surface. It follows that the formal Euler characteristic is deg(α) =−4. The Hom spaces are now_Q[α]-modules, so we have lost information by killing all integer torsion, but having the coefficients invertible allows us to use a variant of Gaussian elimination that is essential in Theorem 4.10.

This can be considered in some ways as a generalisation of Khovanov and Lee homology, since those homology theories can be recovered from CKhα_{: for a link L, the condition that = 0}

implies α= 0 and = 8 implies α= 8. So if we do the calculation for the trefoil for Khovanov homology as we did earlier in Example 2.29, but working over _Q[α], we obtain the complex

BN(31)'q∅ ⊕q3∅ → • →q5∅

α

−→q9∅.

Setting α= 0, the above complex is quasi-isomorphic to the Khovanov complexCKh(31), and

settingα−8 = 0, the above complex is quasi-isomorphic to the Lee complexCKhLee₍₃

1) (since

Q−→8 Qis contractible). Thus,

Kh(L)•∼=H•(CKhα(L)⊗Q[α]/(α= 0))

KhLee(L)•∼=H•(CKhα(L)⊗Q[α]/(α−8 = 0)).

CHAPTER 4. LEE SPECTRAL SEQUENCE4.2. EQUIVARIANT KHOVANOV HOMOLOGY

Note. We could alternatively recover Lee homology by tensoring with Q[α]/(α−1 = 0) by the

homotopy equivalence

CKhα(L)⊗Q[α]/(α−8 = 0)'CKhα(L)⊗Q[α]/(α−1 = 0).

Digression. There is a subtlety involved when working with Hom above. When we take Hom(∅,−), we are not taking the set of chain complex homomorphisms, rather we are taking the ‘internal Hom’ for the category of chain complexes. Explicitly, there are two categories under consideration:

• the categoryKomwhose objects are chain complexes, and whose morphisms are chain maps. The set of morphisms between two complexes forms a vector space.

• the categoryKomd whose objects are again chain complexes, but morphisms are linear maps

f :L

C• →LD•, and these maps form a chain complex. That is, Komd(C•, D•)i ={f :

Cj→Dj+i}with differential (df)(x) =d(f(x)) + (−1)degff(dx). Since Hom(C•, D•) is itself a chain complex, i.e. it is an object in the category again, it is called ‘internal Hom’. This is the same as taking the usual Hom cochain complex in homological algebra. Importantly, for homotopic complexesD•'D0•,

Hom

d

Kom(C•, D•)'HomKomd(C•, D

0 •),

so taking homology will give us information about the original chain complexes.

Example 4.3. In Definition 4.10, we are taking the ‘internal Hom’ (think of∅as the chain complex concentrated in degree zero). Hence, taking HomBN(∅,−) into this complex (which is a covariant

functor, that is only left exact, but that doesn’t matter), we get

CKhα(31) = HomBN(∅, BN(31))'qQ[α]⊕q3Q[α]→ • →q5Q[α]−→α q9Q[α].

This is because HomBN(∅,∅)∼=Q[α] and

HomBN(∅,∅ αn −−→ ∅) = HomBN(∅,∅) d −→HomBN(∅,∅) ∼ =Q[α] α n −−→_Q[α]

where dis multiplication byαn_{, by the rule (df)(x) =d(f(x)) + (−1)}degf_{f(dx) which implies that}

for any polynomialp(α)∈HomBN(∅,∅),dp(α) =αnp(α).

To see that HomBN(∅,∅) ∼= Q[α], we need to show that any Σ ∈ HomBN(∅,∅) ∼= Q[α] can be

reduced, via local relations, to a polynomial function of a genus 3 surface (or a lower genus closed surface, which will be equal to a scalar). Note that Σ is a closed surface. SinceBN is an additive category, HomBN is closed under addition and we can take linear combinations of elements in

HomBN. Thus we just need to show that we can reduce each surface Σ to a monomial inQ[α]. Let

Σ0 be a connected component of Σ and letgdenote genus. Since all closed orientable surfaces are classified up to genus, and cobordisms are orientable, it suffices to check this for different genera. Ifg(Σ0)<3, then Σ0∈ {0,2} by the local relations inCob3_•/l. Ifg(Σ0) = 3, then clearly Σ0= 8αby

Definition 4.10. Now supposeg(Σ) =n >3, and think of this as an genusnsurface

· · ·

then iteratively use the neck cutting relation two holes down from the end (along dotted line) to reduce Σ0 to some monomial in α. Multiplying the results of the connected components of Σ, we obtain a monomial inα.

4.2. EQUIVARIANT KHOVANOV HOMOLOGYCHAPTER 4. LEE SPECTRAL SEQUENCE

Theorem 4.4. The homologyKhα(L)of a link L is a link invariant.

The proof of this is exactly the same as in [BN07]: compute and simplify the complexes corresponding to each side of the Reidemeister move, and one gets the same result for both sides of any given Reidemeister move, and hence we have invariance.

The complexKhα_{(L) is an object in the category of graded complexes of}

Q[α]-modulesGKom(Q[α]−

Mod), for which we have the following Definition and Lemma which gives a version of ‘Gaussian elimination’ (see Proposition 2.5) for_Q[α]-modules, found in [HKLM15].

Definition 4.5. Let C be an additive category. Then C is a Krull-Schmidt category if it is semisimple (i.e. can be written as a finite sum of indecomposable objects) and for any two decompositions into simple objects

M1⊕M2⊕ · · · ⊕Mn=N1⊕N2⊕ · · · ⊕Nn,

there exists a permutationσof{1, . . . , n}such thatNi∼=Mσ(i).

Lemma 4.2. The homotopy category GKom(Q[α]−Mod) of finite length graded complexes of Q[α]-modules is Krull-Schmidt.

Proof. Take a complexC•whose objects are direct sums ofQ[α]{r}, i.e. a polynomial ring over the

formal variableαwithq-grading shifts, and whose differentials have degree zero. Since the maps are degree zero, the differentials must be matrices of monomials. Explicitly, a degree zero map

M j∈J Q[α]{rj} → M k∈K Q[α]{r0k}

is a matrix whose (j, k) entry is a degree zero mapQ[α]{rj} →Q[α]{r0k}. That is, multiplication

byzαrj−r0k _{for some z}∈_Q.

We will first show that we can break downC• into a direct sum of copies of 0→Q[α]→0

and

0→_Q[α]−α−→r _Q[α]→0 forr≥0 at various homological andq-gradings.

If the differentials ofC• are zero, then we are done. Suppose the differentialdn is nonzero (has a

nonzero entry). Take the nonzero entry with lowest degree monomial. By permuting direct sums, we may assume that this entry is the top left entry,

dn = zαrj−r0k δ γ ,

where zαrj−r0k divides each other entry. Becausezαrj−r0k divides each other entry, we can do a

version of Gaussian elimination described in Proposition 2.5. Hence, the following portion of chain complex · · · A Q[α]{r} ⊕B Q[α]{s} ⊕C D · · ·   ξ ζ     φ δ γ   µ ν

CHAPTER 4. LEE SPECTRAL SEQUENCE4.2. EQUIVARIANT KHOVANOV HOMOLOGY Q[α]{r} Q[α]{s} · · · A B C D · · · zβr−s ⊕ ⊕ ζ −γ(z−1_αs−r_)δ ν

Thus we get the desired decomposition. Moreover, the resulting complex is homotopy equivalent to grading shifted copies of

0→Q[α]→0 and 0→Q[α] α r −−→Q[α]→0 forr >0, since 0→Q[α] α 0 −→Q[α]→0

is contractible becauseα0_{is invertible.}

Lastly, to show that the homotopy category of GKom(Q[α]−Mod) is Krull-Schmidt, we need

to show that each decomposition is unique up to homotopy. Suppose we have two homotopic complexes C•=C0⊗(0→Q[α]→0)⊕ M i≥1 Ci⊗ 0→_Q[α] β i −→_Q[α]→0 C_•0 =C₀0 ⊗(0→_Q[α]→0)⊕M i≥1 C_i0⊗ 0→_Q[α] α i −→_Q[α]→0

where Ci andCi0 are doubly graded vector spaces. Tensoring withQ[α]/(α−1), we have

C0∼=H(C⊗Q[α]/(α−1))∼=H(C0⊗Q[α]/(α−1))∼=C00.

Next consider tensoring with_Q[α]/(αr_{). We firstly compute}

H 0→Q[α] α i −→Q[α]→0 ⊗Q[α]/(αr) =H 0→Q[α](αr) α i −→Q[α]/(αr)→0 = ( Qr⊕tQr i≥r Qr−i⊕tQr−i i < r Thus, H(C•⊗Q[α]/(αr)) =C0⊗Qr⊕ M i≥r Ci⊗(1 +t)Qr⊕ M r>i≥1 Ci⊗(1 +t)Qr−i

by using the result of Example 3.8. Hence, formally, we have

H(C•⊗Q[α]/(αr+1))−H(C•⊗Q[α]/(αr)) =C0⊕(1 +t) M i≥r Ci, and thus −H(C•⊗Q[α]/(αr+2)) + 2H(C•⊗Q[α]/(αr+1))−H(C•⊗Q[α]/(αr)) = (1 +t)Cr

which allows us to compute the graded dimensions of the Ci and this is independent of homotopy,

thusCi∼=Ci0.

Remark 4.6. Note that in the above proof, we need zto be invertible, which is why we chose earlier to work over_Q.

4.3. COMPARING WITHKhα _{CHAPTER 4. LEE SPECTRAL SEQUENCE}

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